Nakamura, Jinya

On the Milnor $K$-groups of complete discrete valuation fields

Doc. Math., J. DMV 5, 151-200 (2000)


Summary: For a discrete valuation field $K$, the unit group $K^{\times}$ of $K$ has a natural decreasing filtration with respect to the valuation, and the graded quotients of this filtration are given in terms of the residue field. The Milnor $K$-group $\MK_q(K)$ is a generalization of the unit group, and it also has a natural decreasing filtration. However, if $K$ is of mixed characteristics and has an absolute ramification index greater than one, the graded quotients of this filtration are not yet known except in some special cases.

Mathematics Subject Classification

19D45, 11S70


the Milnor $K$-group, complete discrete valuation field, higher local class field theory